Pith Number

pith:N3TLD6PR

pith:2026:N3TLD6PRU5UUFIUMZBO4X3D7AC

not attested not anchored not stored refs pending

Admissible subgraphs and the depth of symbolic powers of cover ideals of graphs

Nguyen Thu Hang, Thanh Vu, Tran Duc Dung

The depth of the t-th symbolic power of the cover ideal of a cycle graph C_n equals n-1 minus floor of t n over 2t plus 1.

arxiv:2605.03369 v2 · 2026-05-05 · math.AC

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\pithnumber{N3TLD6PRU5UUFIUMZBO4X3D7AC}

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Claims

C1strongest claim

We prove that depth(S/J(C_n)^{(t)}) = n - 1 - floor(tn/(2t+1)) for all t ≥ 2 and n ≥ 3, where S = K[x1,...,xn] and J(C_n) is the cover ideal of the cycle on n vertices.

C2weakest assumption

That the newly defined t-admissible subgraphs correctly encode the depth information for the symbolic powers, allowing the reduction to the stated formula for cycles without hidden restrictions on the graph or the base field.

C3one line summary

The depth of the t-th symbolic power of the cover ideal of the cycle graph C_n equals n-1 minus the floor of tn over 2t+1, for t at least 2 and n at least 3.

Cited by

1 paper in Pith

Critical subgraphs and the regularity of symbolic powers of cover ideals of graphs

Receipt and verification

First computed	2026-05-20T01:05:15.280579Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

6ee6b1f9f1a76942a28cc85dcbec7f00902a698c32de73fc1c56820ad8fd90f2

Aliases

arxiv: 2605.03369 · arxiv_version: 2605.03369v2 · doi: 10.48550/arxiv.2605.03369 · pith_short_12: N3TLD6PRU5UU · pith_short_16: N3TLD6PRU5UUFIUM · pith_short_8: N3TLD6PR

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/N3TLD6PRU5UUFIUMZBO4X3D7AC \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 6ee6b1f9f1a76942a28cc85dcbec7f00902a698c32de73fc1c56820ad8fd90f2

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "9ca86d3b0cca3c946872f0ca4af299ac811453ac1b12eca074dec4ae3f537ba7",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
    "primary_cat": "math.AC",
    "submitted_at": "2026-05-05T05:19:06Z",
    "title_canon_sha256": "f6b7c1b69e6045214e5ce53c56c4b723424c9e036cee91da4f2f913080d2f93e"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.03369",
    "kind": "arxiv",
    "version": 2
  }
}